System and device including reconfigurable physical unclonable functions and threshold cryptography

ABSTRACT

A system and device, including reconfigurable physical unclonable functions (‘RPUFs’) and threshold cryptography, use cryptographic and physical means of security. A plurality of reconfigurable physical unclonable functions (‘RPUFs’) and a memory are connected to a processor that is configured to derive information associating the RPUFs with cryptographic shares of a sensitive value, store such information in the memory, and reconfigure a RPUF upon powering up of the device such that information stored in the memory is not valid for the reconfigured RPUF.

FIELD OF THE DISCLOSURE

This disclosure relates generally to hardware verification, and in particular but not exclusively, to binding authentication to protect against tampering and subversion by substitution.

BACKGROUND

A physical unclonable function or ‘PUF’ is a physical entity capable of generating an output (‘response’) to a given input (‘challenge’) that is unique to that particular PUF such that it can be regarded as a ‘fingerprint.’ This capacity is typically arrived at by devising the PUF in such a way that its output depends upon features that differ randomly in each device due to minor manufacturing variations. Thus a PUF cannot be readily replicated with the correct fingerprint, even with full knowledge of its circuit layout.

Typical PUF-based protocols fall into two broad categories: (1) a simple challenge-response provisioning process like the one described below in Protocol 1, or (2) cryptographic augmentation of a device's PUF response such that the raw PUF output never leaves the device. These approaches may require external entities to handle auxiliary information (e.g., challenges and their associated helper data) that is unsupported or superfluous in existing public key cryptography standards, and/or involve a hardware device authenticating to a challenge applied during an initial enrollment process, and/or are premised on the hardware device always recovering essentially the same response to a given challenge.

One challenge facing PUF-based systems is side channel attacks, which seek to observe and analyze auxiliary environmental variables to deduce information about sensitive information, such as cryptographic key material or the PUF output.

A reconfigurable PUF or ‘RPUF’ is a PUF that can be altered to generate a new set of responses to the same challenges. A PUF's set of responses and challenges is often called a ‘mapping,’ and the new mapping after reconfiguration of a RPUF is preferably fully different from and unrelated to the RPUF's other challenge-response mappings. RPUFs can be reversibly configurable such that after reconfiguration the RPUF can be reconfigured again to a prior configuration, or irreversibly configurable such that it is not possible to restore a prior challenge-response mapping. Logically-reconfigurable PUFs (‘L-RPUFs’) utilize a programmable feature to change configurations and are typically reversible, and physically-reconfigurable PUFs (‘P-RPUFs’) are reconfigured by imposing a physical change and are typically irreversibly configurable.

SUMMARY OF THE DISCLOSURE

A system and device including reconfigurable physical unclonable functions (‘RPUFs’) and threshold cryptography uses cryptographic and physical means of security. In one embodiment, a plurality of reconfigurable physical unclonable functions (‘RPUFs’) and a memory are connected to a processor that is configured to derive information associating the RPUFs with cryptographic shares of a sensitive value, store such information in the memory, and reconfigure a RPUF upon powering up of the device such that information stored in the memory is not valid for the reconfigured RPUF. One or more redundant RPUFs may be used in place of one PUF. When the system or device is powered up, one of the RPUFs is selected and transitioned from its previous configuration to a new configuration, invalidating any information associating the RPUF with shares of a sensitive value based on the old state of that RPUF. An RPUF that was not reconfigured is then used to recover the reconfigured RPUF's shares using its stored information.

Further embodiments provide for shares to be held by multiple PUFs so that each PUF is mapped to less than all shares. These embodiments can provide power failure event resiliency, in contrast to an approach where individual PUFs are associated with all shares. (t, n) threshold sharing can be tailored so t is less than n (e.g., by 1, 2, 3, etc.) to enable tolerance for benign power events.

Alternately, one RPUF may be used for each share of the sensitive value. An RPUF is selected and transitioned from its previous configuration to a new configuration when the device is powered up, invalidating any information associating the RPUF with a share of a sensitive value based on the old state of that RPUF. RPUFs that were not reconfigured can then be used to generate a replacement share for the reconfigured RPUF's invalidated share using its stored information.

In another embodiment based on a single PUF queried with multiple challenges (to generate different shares of the sensitive value), an RPUF is used with one parameter to recover values and a different parameter to store values, with the values not being repeated in ensuing share-refresh cycles. In another embodiment, a pair of RPUFs is used instead, with one RPUF used to recover values and another RPUF used to store values.

According to one aspect, an authenticatable device, implementing a threshold sharing of a hardware based identity that comprises a (t, n) sharing of a sensitive value, is provided. The authenticatable device comprises n reconfigurable physical unclonable functions (‘RPUFs’) associated with at least one respective share of the threshold sharing of the sensitive value, a memory, and at least one processor configured to: invalidate any shares of a respective RPUF based on reconfiguration of the respective RPUF, reconfigure, randomly, one of the n RPUFs responsive to power-up of the device, and generate information derived from at least t valid RPUF shares to enable one or more cryptographic operations that require the sensitive value without generating the sensitive value in memory.

According to one aspect, an authenticatable device implementing a threshold sharing of a hardware based identity that comprises a (t, n) sharing of a sensitive value is provided. The authenticatable device comprises n reconfigurable physical unclonable functions (‘RPUFs’) associated with at least one respective share of the threshold sharing of the sensitive value, a memory, and at least one processor configured to: reconfigure, randomly, one of the n RPUFs after power-up of the device invalidating any share of the reconfigured RPUF, and perform one or more cryptographic operations that require the sensitive value without generating the sensitive value, utilizing information derived from at least t valid RPUF shares.

According to one embodiment, the at least one processor is further configured to identify invalid shares. According to one embodiment, the at least one processor is further configured to construct a valid replacement share for a reconfigured RPUF, based on at least t valid RPUF shares. According to one embodiment, the at least one processor is configured to store a challenge and helper value for the valid replacement share. According to one embodiment, the at least one processor is configured to enroll the n RPUFs in the (t, n) sharing of the sensitive value without generating the sensitive value in the memory. According to one embodiment, the n RPUFs are associated with the at least one respective share of the sensitive value by an associated challenge and helper pair.

According to one embodiment, the at least one processor is configured to enroll the n RPUFs in the (t, n) sharing of the sensitive value so that t valid shares are required to reconstruct the sensitive value where t is less than n. According to one embodiment, the threshold sharing is constructed where n−t is at least two. According to one embodiment, the n RPUFs are associated with multiple shares. According to one embodiment, the at least one processor is configured to trigger a share refresh responsive to enabling a request to execute the one or more cryptographic operations.

According to one embodiment, the at least one processor is configured to generate sub-shares associated with the valid RPUF shares. According to one embodiment, the at least one processor is configured to blind the sub-shares. According to one embodiment, the at least one processor is configured to generate threshold shares of zero values to blind the sub-shares.

According to one aspect, a computer implemented method for threshold sharing of a hardware based identity comprising a (t, n) sharing of a sensitive value on an authentication device is provided. The method comprises: randomly reconfiguring, by at least one processor, one of n reconfigurable physical unclonable functions (‘RPUFs’) responsive to power-up of the device invalidating any share of the reconfigured RPUF, issuing, by the at least one processor, challenges to at least t RPUFs to recover respective shares of the threshold sharing of the sensitive value, and enabling, by the at least one processor, one or more cryptographic operations that require the sensitive value without generating the sensitive value in memory utilizing information derived from at least t valid RPUF shares.

According to one embodiment, the method further comprises combining a threshold operation on multiple shares to generate an output that enables the one or more cryptographic operations. According to one embodiment, the method further comprises communicating the output for execution of the one or more cryptographic operations.

According to one embodiment, the method further comprises identifying, by the at least one processor, invalid shares. According to one embodiment, the method further comprises constructing, by the at least one processor, a valid replacement share for an invalid share based on at least t valid RPUF shares. According to one embodiment, the method further comprises storing, by the at least one processor, a challenge and helper pair for the valid replacement share.

According to one embodiment, the method further comprises enrolling the n RPUFs in the (t, n) sharing of the sensitive value without generating the sensitive value in the memory. According to one embodiment, the method further comprises associating, by the at least one processor, the n RPUFs with at least one respective share of the sensitive value with an associated challenge and helper pair. According to one embodiment, the method further comprises triggering, by the at least one processor, a share refresh responsive to enabling a request to execute the one or more cryptographic operations.

According to one aspect, an authenticatable device implementing a threshold sharing of a hardware based identity that comprises a (t, n) sharing of a sensitive value is provided. The authenticatable device comprises n reconfigurable hardware identity circuits for generating respective hardware specific output based on a respective input challenge, the n reconfigurable hardware identity circuits associated with the at least one respective share of the threshold of the sensitive value, a memory, and at least one processor configured to: reconfigure, randomly, one of the n hardware identity circuits after power-up of the device invalidating any share of the reconfigured hardware identity circuit, and perform one or more cryptographic operations that require the sensitive value without generating the sensitive value, utilizing information derived from at least t valid hardware identity circuit shares. According to one embodiment, the n hardware identity circuits comprise reconfigurable physical unclonable functions (‘RPUFs’).

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a functional diagram of a device having a single PUF circuit and two threshold shares;

FIG. 2 is a functional diagram of a device having dual PUF circuits;

FIG. 3 is an operational flowchart of enrollment of a device like that of FIG. 2 in an embodiment of the invention;

FIG. 4 is an operational flowchart of threshold cryptographic operations in a device like that of FIG. 2 in an embodiment of the invention;

FIG. 5 is an operational flowchart of the staggered threshold cryptographic operations in a device like that of FIG. 1 in an embodiment of the invention;

FIG. 6 is an operational flowchart of threshold cryptographic operations in a device having redundant reconfigurable PUFs;

FIG. 7 is an operational flowchart of power-on PUF reconfiguration and replacement share construction in a device using one reconfigurable PUF per share; and

FIG. 8 is an operational flowchart of threshold cryptographic operations in a device using one reconfigurable PUF per share.

DETAILED DESCRIPTION

According to some embodiments, compliance with share expiration protocols can be enforced through the use of reconfigurable PUFs (RPUFs). In one embodiment of the invention, a device's PUF root of trust's challenge-response behavior is internalized and used to generate shares of a sensitive value (e.g., a private key) over which shares the device can execute arbitrary threshold cryptographic operations (e.g., decryption, digital signature generation, zero knowledge proofs). Preferably the threshold operations can be executed without ever generating, reconstructing, or storing the sensitive value. In further embodiments, the shares are periodically refreshed (updated) and threshold share operations may be staggered such that one share is stored as a challenge-helper pair, and thus only one share exists in volatile memory at any given time. In one embodiment, multiple PUFs are used to generate multiple corresponding shares; in another embodiment, a single PUF is queried multiple times with different challenges to generate multiple corresponding shares. RPUFs can be used in place of ordinary PUFs to increase the security of this scheme.

Aspects of the disclosure are described with reference to the example of an embodiment utilizing elliptic curve cryptography (including the associated terminology and conventions), but the described embodiments and approaches apply equally to various other cryptographic schemes such as ones employing different problems like discrete logarithm or factoring (in which regard the teachings of U.S. Pat. No. 8,918,647 are incorporated here by reference), and the disclosure is not limited by the various additional features described herein that may be employed with or by virtue of providing examples of implementation.

The following detailed description sets forth details of various embodiments concerning any one or more of the following: 1) threshold cryptography, wherein a sensitive value is divided into shares; 2) the use of PUFs with cryptographic shares; 3) the periodic adoption of new (‘refreshed’ or ‘updated’) shares; 4) dynamic membership (i.e., removal or addition) of shares in threshold cryptography; and 5) embodiments employing reconfigurable PUFs to enforce the invalidation of shares.

Threshold Cryptography

Threshold cryptography involves distributing cryptographic operations among a set of participants such that operations use the collaboration of a quorum of participants. In some conventional approaches, trusted dealer

generates a master asymmetric key pair

^(pub), p^(priv)

for the set of participants p_(i)∈

, |

|=n. The private key is then split among the n participants, with each participant receiving a share of

^(priv). This constitutes a (t, n) sharing of

^(priv), such that a quorum of at least t participants combine their private shares in order to perform operations using the master private key.

While other secret schemes can be used, an example will be described employing Shamir's polynomial interpolation construction, which can be used for sharing a secret. A polynomial ƒ(⋅) of degree t−1 is defined, where the coefficients c_(i) remain private: ƒ(x)=c₀+c₁x+ . . . +c_(t−1) mod q. Without knowledge of the coefficients, ƒ(⋅) can be evaluated when at least t points of ƒ(⋅) are known by applying Lagrange's polynomial interpolation approach. A private key

^(priv) can be set as the free coefficient c₀ (i.e., ƒ(0)=

^(priv)), and a set of shares of the private key distributed to the participants. To split the private key

^(priv) among n participants p₁∈

_(1≤i≤n), the dealer computes p_(i)'s (public, private) key pair as

r_(i)·G mod q, r_(i)

such that r_(i)=ƒ(i), i≠0. Here, G∈E/

_(p) is a base point of order q for elliptic curve E, and (P)_(x) (resp. (P)_(y)) refers to the x (resp. y) coordinate of point P on curve E. (The modulus that operations are performed under may be omitted where it is apparent from context). The public keys are made available to all participants, while the private keys are distributed securely to each participant (e.g., using the device's public key and ElGamal encryption). All participants are also given access to (c_(j)·G)_(0≤j≤t−1), which allows them to verify their secret key and the public keys of other participants by checking that:

${r_{i} \cdot G} = {\sum\limits_{j = 0}^{t - 1}{{i^{j}\left( {c_{j} \cdot G} \right)}\;{mod}\; p}}$ This constitutes a (t, n) verifiable secret sharing (VSS) of the private key

^(priv), as participants are able to verify the legitimacy of their share with respect to a globally-known public key.

Now, given access to any t shares {(i, r_(i))}_(1≤i≤t), where ƒ(⋅) has degree t−1 and t≤n, the shares (i, r_(i)) may be combined through Lagrange polynomial interpolation to evaluate ƒ(x):

${f(x)} = {\sum\limits_{i = 1}^{t}{\left( {r_{i} \cdot \left( {\prod\limits_{{j = 1}{j \neq i}}^{t}\frac{x - x_{j}}{x_{i} - x_{j}}} \right)} \right)\;{mod}\; q}}$ This allows any quorum of t participants p_(i)∈

⊆

, |

⊕=t≤n to combine their shares {(i, r_(i))}_(1≤i≤t) and recover the polynomial's free coefficient c₀=ƒ(0), which is the master asymmetric private key

^(priv). Although the Lagrange form is used for the interpolating polynomial, other approaches (e.g., using a monomial basis or the Newton form) may be substituted. Similarly, although the exemplary construction evaluates ƒ(⋅) rather than recover the coefficients, alternatively the latter may be accomplished using a Vandermonde matrix representation and solving the system of linear equations.

Although elliptic curve cryptography may be used, it will be readily apparent that various other cryptographic frameworks (e.g., EIGamal, RSA, NTRU, etc.) could be employed instead. A number of threshold cryptographic operations can be carried out within this framework, using a variety of methods such as threshold encryption, decryption, and signing, threshold zero knowledge proofs of knowledge, threshold signcryption, and distributed key generation. Other elliptic curve mechanisms such as Massey-Omura, Diffie-Hellman, Menezes-Vanstone, Koyama-Maurer-Okamoto-Vanstone, Ertaul, Demytko, etc. could likewise be employed.

An entity in possession of a device's enrollment information {p_(i) ^(pub), c_(i), helper_(i)} can thus encrypt a message nt such that the target device is able to recover it, using a method such as ElGamal encryption:

ElGamal Encryption for Server s do  Lookup: challenge c_(i), generator G, modulus p and Helper h_(i) for  Device p_(i)  Select y ∈ 

 _(p) uniformly at random  Lookup public key A = p_(i) ^(priv) · G mod p for Device p_(i)  Generate shared key: y · G mod p  Encrypt message m by computing m + (yA)_(y) mod q  Device p_(i) ← {yG, m + (yA)_(y) mod q, c_(i), G, p, h_(i)} end for

Then, if all participants of a group

⊆

, where |

|≥t, |

|=n and t≤n, wish to decrypt an encryption (yG, m+(yrG)_(z)) of a message m∈[1, p−1] using group private key r, threshold ElGamal decryption (e.g., per Ertaul) can be used as follows:

-   -   Each participant p_(i)∈         uses their secret key r_(i)=ƒ(i) to compute a shadow:

$w_{i} = {{\left( {\prod\limits_{{j = 1}{j \neq i}}^{t}\frac{- j}{i - j}} \right) \cdot r_{i}}\;{mod}\; q}$

-   -   Each participant then broadcasts their partial decryption S_(i)         defined as S_(i)=w_(i)·yG mod q.     -   Each participant locally computes the value:

$\begin{matrix} {S = {\sum\limits_{i = 1}^{t}{S_{i}{mod}\; q}}} \\ {= {\left( {\sum\limits_{i = 1}^{t}w_{i}} \right) \cdot {yG}}} \\ {= {r \cdot {yG}}} \end{matrix}$

-   -   Finally, each participant may now locally recover the message m         by computing (m+(yrG)_(y))−S mod q=(m+(yrG)_(y))−(ryG)_(y)=m.

Likewise, a group

⊆

where |

|≥t, |

|=n and t≤n, can use a threshold signature scheme to generate a signature representing all of

for message m as follows:

-   -   Each participant p_(i)∈         uses their secret key r_(i)=ƒ(i) and a random integer y_(i)∈         _(q) to calculate their individual signature (R_(i), S_(i)) for         message m.         -   First, R_(i) is computed from y_(i)·G mod p and publicized             to all participants p_(i)∈             .         -   Next, each participant p_(i) calculates R, e, and S_(i) as             follows:

$R = {\sum\limits_{i = 1}^{t}{R_{i}{mod}\; p}}$ e = h(m, (R)_(y)mod q) $S_{i} = {y_{i} + {r_{i}{e\left( {\prod\limits_{{j = 1}{j \neq i}}^{t}\frac{- j}{i - j}} \right)}{mod}\; q}}$

-   -   -   h(⋅) or H(⋅) denotes a cryptographic hash function. Each             participant broadcasts S_(i) to an appointed secretary (for             convenience, and who need not be trusted).

    -   The secretary, having received all (R_(i), S_(i)) pairs,         verifies the signature by computing:

$R = {\sum\limits_{i = 1}^{t}{R_{i}{mod}\; p}}$ e = h(m, (R)_(y)mod q) $R_{i} = {{S_{i} \cdot G} + {{e\left( {\prod\limits_{{j = 1}{j \neq i}}^{t}\frac{- j}{i - j}} \right)}\left( {- p_{i}^{pub}} \right){mod}\; p}}$

If constructed properly, this equation will hold as:

$\begin{matrix} {R_{i}\overset{?}{=}{{S_{i} \cdot G} + {{e\left( {\prod\limits_{{j = 1}{j \neq i}}^{t}\frac{- j}{i - j}} \right)}\left( {- p_{i}^{pub}} \right){mod}\; p}}} \\ {= {{\left( {y_{i} + {r_{i}{e\left( {\prod\limits_{{j = 1}{j \neq i}}^{t}\frac{- j}{i - j}} \right)}}} \right) \cdot G} + {{e\left( {\prod\limits_{{j = 1}{j \neq i}}^{t}\frac{- j}{i - j}} \right)}\left( {{- r_{i}}G} \right)}}} \\ {= {{y_{i}G} + {{er}_{i}{G\left( {\prod\limits_{{j = 1}{j \neq i}}^{t}\frac{- j}{i - j}} \right)}} + {{e\left( {{- r_{i}}G} \right)}\left( {\prod\limits_{{j = 1}{j \neq i}}^{t}\frac{- j}{i - j}} \right)}}} \\ {= {{y_{i}G} + {\left( {{{er}_{i}G} - {{er}_{i}G}} \right)\left( {\prod\limits_{{j = 1}{j \neq i}}^{t}\frac{- j}{i - j}} \right)}}} \\ {= {y_{i}G}} \\ {:=R_{i}} \end{matrix}$

-   -   If these hold, the secretary calculates:

$S = {\sum\limits_{i = 1}^{t}{S_{i}{mod}\; q}}$ which computes the group signature ((R)_(y) mod q, S) over m.

-   -   Upon receipt of (R, S), receiver p_(R) checks its validity         against the public key         ^(pub) of the entire group of participants p_(i)∈         _(1≤i≤n);         e         h(m,(S·G+r·−         ^(pub))_(y) mod q)     -   which holds on valid signatures because:

$\begin{matrix} {e\overset{?}{=}{h\left( {m,{\left( {{S \cdot G} + {e \cdot {- \mathcal{P}^{pub}}}} \right)_{y}{mod}\; q}} \right)}} \\ {= {h\left( {m,\left( {{\left( {y + {re}} \right) \cdot G} + {e \cdot \left( {- {rG}} \right)}} \right)_{y}} \right)}} \\ {= {h\left( {m,\left( {{yG} + {erG} - {erG}} \right)_{y}} \right)}} \\ {= {h\left( {m,({yG})_{y}} \right)}} \\ {= {h\left( {m,{(R)_{y}{mod}\; q}} \right)}} \end{matrix}$

The participants of a group

⊆

where

|≥t, |

|=n and t≤n can also collaborate to demonstrate possession of a shared private key

^(priv)=r∈[1, q−1] using a threshold Zero Knowledge Proof of Knowledge (e.g., Sardar et al., “Zero Knowledge Proof in Secret Sharing Scheme Using Elliptic Curve Cryptography,” Global Trends in Computing and Communication Systems, volume 269 of Communications in Computer and Information Science, pages 220-226, Springer, 2012) as follows:

-   -   The group public key is         ^(pub)=rG, where r is a shared secret and G is a group         generator. The verifier         chooses an ephemeral nonce N and distributes this to all         participants of         .     -   Each participant p_(i)∈         uses their secret share r_(i)=ƒ(i) and a random nonce integer         y_(i) to calculate their individual proof (B_(i), M_(i)) of the         shared secret r.         -   First, B_(i) is computed and publicized to all participants             p_(i)∈             :             B _(i) =y _(i) ·G mod p         -   Each participant locally computes:

$B = {{y \cdot G} = {\sum\limits_{i = 1}^{t}{B_{i}{mod}\; p}}}$

-   -   -   Next, each participant p_(i) calculates e, M_(i) as follows:

e = h(G, B, 𝒫^(pub), N) $M_{i} = {y_{i} + {r_{i}{e\left( {\prod\limits_{{j = 1}{j \neq i}}^{t}\frac{- j}{i - j}} \right)}{mod}\; q}}$

-   -   Upon receipt of (B_(i), M_(i))_(1≤i≤t), the verifier         computes:

$B = {\sum\limits_{i = 1}^{t}{B_{i}{mod}\; p}}$ $M = {\sum\limits_{i = 1}^{t}{M_{i}{mod}\; q}}$ e = h(G, B, 𝒫^(pub), N)

-   -   Next, the verifier checks the proofs validity against the public         key         ^(pub)=rG.

$\begin{matrix} {B\overset{?}{=}{{M \cdot G} - {{e \cdot \mathcal{P}^{pub}}{mod}\; p}}} \\ {= {{\left( {y + {re}} \right) \cdot G} - {e \cdot ({rG})}}} \\ {= {{yG} + {reG} - {reG}}} \\ {= {yG}} \end{matrix}$ If B=M·G−e·

^(pub), the verifier

accepts the threshold zero knowledge proof as valid, and rejects the proof otherwise.

The process of signcrypting (e.g., Changgen et al., “Threshold Signcryption Scheme based on Elliptic Curve Cryptosystem and Verifiable Secret Sharing,” International Conference on Wireless Communications, Networking and Mobile Computing, volume 2, pages 1182-1185, 2005; Zheng, “Digital Signcryption or How to Achieve Cost(Signature & Encryption)«Cost(Signature)+Cost(Encryption),” Advances in Cryptology, CRYPTO '97, volume 1294 of Lecture Notes in Computer Science, pages 165-179, Springer, 1997; Zheng et al., “How to Construct Efficient Signcryption Schemes on Elliptic Curves,” Inf. Process. Lett., volume 68, no. 5:227-233, 1998) a message facilitates performing both signing and encrypting a message at a cost less than computing each separately. Given a message m∈[1, q−1] and a receiver p_(R) with public key p_(R) ^(pub), signcryption can be generated as follows:

-   -   Each p_(i)∈         selects a random k_(i)∈[1, q−1] and computes Y_(i)=k_(i)·G and         publicly broadcasts this to both a secretary (for convenience,         and who need not be trusted) and the receiver p_(R). Each p_(i)∈         also computes Z_(i)=k_(i)·p_(R) ^(pub) which is privately (for         example, using ElGamal encryption) sent to p_(R).     -   The secretary computes:

$Z = {{\sum\limits_{i = 1}^{t}Z_{i}} = {{\sum\limits_{i = 1}^{t}{k_{i} \cdot p_{R}^{pub}}} = {k \cdot p_{R}^{pub}}}}$ r = m ⋅ (Z)_(x) mod p

-   -   and broadcasts r (not to be confused with r_(i), participant         p_(i)'s share of         ^(priv)) to each signer p_(1≤i≤t).     -   Each signer p_(1≤i≤t) computes:

$x_{i} = {\prod\limits_{\underset{j \neq i}{j = 1}}^{t}{\frac{- j}{i - j}\;{mod}\; q}}$ e_(i) = r_(i) ⋅ x_(i) mod q s_(i) = k_(i) − e_(i) ⋅ r  mod q

-   -   where r_(i)=ƒ(i) is p_(i)'s share of         ^(priv). Each signer sends their partial signcryption s_(i) to         the secretary.     -   Upon receipt of a partial signcryption s_(i), the secretary         computes Y′_(i)=r·x_(i)·p_(i) ^(pub)+s_(i)·G in order to verify         the validity of the partial signcryption by checking Y_(i)         Y′_(i).     -   Upon receipt of all partial signcryptions s_(i) and checking         their validity, the secretary combines them to compute:

$s = {\sum\limits_{i = 1}^{t}{s_{i}\;{mod}\; q}}$

-   -   and (r, s) is the final signcryption sent to receiver p_(R).     -   The receiving participant p_(R), which has now received         {Y_(i)=k_(i)·G}_(i∈[1 . . . n]), (r, s)         , computes:

$Y = {{\sum\limits_{i = 1}^{t}Y_{i}} = {{\sum\limits_{i = 1}^{t}\left( {k_{i} \cdot G} \right)} = {k \cdot G}}}$ Y^(′) = r ⋅ 𝒫^(pub) + s ⋅ G Z^(′) = p_(R)^(priv) ⋅ Y^(′)

-   -   The recipient p_(R) then verifies that:

$\begin{matrix} {{Y^{\prime}\overset{?}{=}r}{{\cdot \mathcal{P}^{pub}} + {s \cdot G}}} \\ {= {{r \cdot \mathcal{P}^{pub}} + {\sum\limits_{i = 1}^{t}{s_{i} \cdot G}}}} \\ {= {{r \cdot \mathcal{P}^{pub}} + {G \cdot {\sum\limits_{i = 1}^{t}\left( {k_{i} - {e_{i} \cdot r}} \right)}}}} \\ {= {{r \cdot \mathcal{P}^{pub}} + \left( {G \cdot {\sum\limits_{i = 1}^{t}k_{i}}} \right) - \left( {G \cdot {\sum\limits_{i = 1}^{t}{e_{i} \cdot r}}} \right)}} \\ {= {{r \cdot \mathcal{P}^{pub}} + {k \cdot G} - {r \cdot G \cdot {\sum\limits_{i = 1}^{t}e_{i}}}}} \\ {= {{r \cdot \mathcal{P}^{pub}} + {k \cdot G} - {r \cdot G \cdot \left( {\sum\limits_{i = 1}^{t}{r_{i} \cdot \left( {\prod\limits_{\underset{j \neq i}{j = 1}}^{t}\frac{- j}{i - j}} \right)}} \right)}}} \\ {= {{r \cdot \mathcal{P}^{pub}} + {k \cdot G} - {r \cdot G \cdot \left( {f(0)} \right)}}} \\ {= {{r \cdot \mathcal{P}^{priv} \cdot G} + {k \cdot G} - {r \cdot G \cdot \mathcal{P}^{priv}}}} \\ {= {{k \cdot G} = Y}} \end{matrix}\;$

If these hold, the group signature over m, is valid.

-   -   The recipient p_(R) can now recover the message m, by computing:

$\begin{matrix} {{r \cdot \left( Z^{\prime} \right)_{x}^{- 1}} = {\left( {m \cdot (Z)_{x}} \right) \cdot \left( Z^{\prime} \right)_{x}^{- 1}}} \\ {= {\left( {m \cdot \left( {k \cdot p_{R}^{pub}} \right)_{x}} \right) \cdot \left( {p_{R}^{priv} \cdot Y^{\prime}} \right)_{x}^{- 1}}} \\ {= {\left( {m \cdot \left( {k \cdot \left( {p_{R}^{priv} \cdot G} \right)} \right)_{x}} \right) \cdot \left( {p_{R}^{priv} \cdot \left( {k \cdot G} \right)} \right)_{x}^{- 1}}} \\ {= m} \end{matrix}$ With this, the recipient p_(R) has both verified the group's signature over message m, as well as decrypted m. Distributed Key Generation

Standard threshold cryptographic operations (e.g., those discussed above) traditionally use a trusted dealer

to define a generating polynomial ƒ(⋅), select a secret r, and distribute shares of r to all participants p_(i)∈

. Distributed key generation protocols remove the necessity of a trusted dealer, and allow a set of participants

to generate shares of a secret where no one knows the shared secret r. According to one embodiment, this can be accomplished in the present context as follows:

-   -   Each participant p_(i)∈         defines a random polynomial ƒ_(i)(⋅) of degree t−1, where t is         the threshold. The temporary private value of participant p_(i)         is c₀ ^((t)), the free coefficient of ƒ_(i)(⋅).     -   Each participant p_(i)∈         privately sends ƒ_(i)(j) to participant p_(i), ∀j∈[1, . . . ,         n]/i.     -   Participant p_(i) broadcasts {c_(k) ^((i))·G mod         p}_(k∈[0, . . . , t−1]), commitments to the coefficients of         ƒ_(i)(⋅).     -   Participant p_(i) broadcasts {ƒ_(i)(j)·G mod         p}_(j∈[0, . . . , n]), the public shares for all participants.     -   Each participant p_(j≠i)∈         must now verify the shares they have received.         -   First, each participant p_(j≠i) verifies that:

$\begin{matrix} {{{f_{i}(j)} \cdot G}\overset{?}{=}{\sum\limits_{k = 0}^{t - 1}{{j^{k}\left( {c_{k}^{(i)} \cdot G} \right)}\mspace{11mu}{mod}\; p}}} \\ {= {G \cdot \left( {\sum\limits_{k = 0}^{t - 1}{j^{k}c_{k}^{(i)}}} \right)}} \\ {= {G \cdot {f_{i}(j)}}} \end{matrix}$

-   -   -   Similarly, each participant p_(j≠i)∈             verifies that their share is consistent with other shares:

$\begin{matrix} {{c_{0}^{(i)} \cdot G}\overset{?}{=}{\sum\limits_{i = 1}^{t}{\left( {\left( {\prod\limits_{\underset{j \neq i}{j = 1}}^{t}\frac{- j}{i - j}} \right){{f_{i}(j)} \cdot G}} \right)\;{mod}\; p}}} \\ {= {G \cdot {\sum\limits_{i = 1}^{t}\left( {\left( {\prod\limits_{\underset{j \neq i}{j = 1}}^{t}\frac{- j}{i - j}} \right){f_{i}(j)}} \right)}}} \\ {= {G \cdot c_{0}^{(i)}}} \end{matrix}$

-   -   If these two verifications succeed, each participant p_(i)∈         computes its share of the master asymmetric private key r:

$r_{i} = {\sum\limits_{j = 1}^{n}{{f_{j}(i)}\mspace{11mu}{mod}\; q}}$

-   -   Similarly, the master asymmetric public key for the group is         computed as:

${r \cdot G} = {\sum\limits_{j = 1}^{n}{\sum\limits_{i = 1}^{n}{\left( {{f_{j}(i)} \cdot G} \right)\;{mod}\; p}}}$

The distributed key generation protocol is preferably secure again an adversary that attempts to bias the output distribution, as in the attack described by Gennaro et al, “Secure Distributed Key Generation for Discrete-Log Based Cryptosystems.” Similarly, threshold constructions are preferably secure against both static as well as adaptive malicious adversaries.

PUF-Enabled Threshold Cryptography

The core functionality of a PUF is extracting a unique mapping between the challenge (input) domain and the response (output) range. As the mapping from challenges to responses is unique for each PUF-enabled device, collecting a set of challenge-response pairs (CRPs) through a provisioning process allows the device to be verified in the future. Protocol 1 illustrates the naïve provisioning process that underlies many PUF-enabled protocols.

Protocol 1 Challenge-Response Provisioning PUF Device D Server s ← challenge c ∈ {0, 1}^(κ) ¹ P(c) 

 r ∈ {0, 1}^(κ) ² → store(D, {c, r}) Authentication proceeds by issuing a challenge for which the response is known to the server, and verifying that the response is t-close to the expected response. However, this lightweight nave protocol has many limitations. During enrollment, a large number of challenge-response pairs must be collected, as each pair can only be used once for authentication. If an adversary observed the response, it could masquerade as the device. Similarly, the challenge-response database is sensitive, as an adversary could apply machine learning to fully characterize the PUF mapping “Modeling Attacks on Physical Unclonable Functions”. These issues can be eliminated by applying cryptographic constructs around the PUF functionality.

In the example of an embodiment employing elliptic curve cryptography, Algorithms 1 and 2 below can be used to allow a PUF-enabled device to locally store and retrieve a sensitive value without needing to store any sensitive information in non-volatile memory. Algorithm 1 illustrates the storing of a sensitive value

_(i) using a PUF, and Algorithm 2 illustrates the dynamic regeneration of

_(t). The challenge c_(i) and helper data helper, can be public, as neither reveals anything about the sensitive value

. While the present example uses encryption of

_(i) by exclusive-or, ⊕,

_(i) could also be used as a key to other encryption algorithms (e.g., AES) to enable storage and retrieval of arbitrarily sized values.

Algorithm 1 PUF-Store Goal: Store value 

 _(i) for PUF Device d do  Select finite field 

 _(n) of order n  Select E, an elliptic curve over 

 _(n)  Find G ∈ E/ 

 _(n), a group generator  Select challenge c_(i) ∈ 

 _(n)  x = H(c_(i), E, G, n)  O = PUF(x)  helper_(i) = P_(i) = O ⊕ ECC( 

 _(i))  Write {c_(i), helper_(i)} to non-volatile memory end for

Algorithm 2 PUF-Retrieve Goal: Retrieve value  

_( i) for PUF Device d do  Read {c_(i), helper_(i)} from non-volatile memory  x ← H(c_(i), E, G, n)  O′ = PUF(x)  

_( i) ← D((ECC( 

_( i)) ⊕ O) ⊕ O′) end for Whenever O and O′ are t-close, the error correcting code ECC can be passed to a decoding algorithm D to recover the sensitive value

_(i).

Using Algorithm 3, a local device can perform an enrollment protocol using the PUF. This allows each PUF circuit to generate a local public key p_(i) ^(pub), which is useful for bootstrapping more complex key setup algorithms (e.g., the distributed key generation protocol in Algorithm 4). When the key setup algorithm is performed internal to the device (rather than externally among a set of distinct devices), this bootstrap process may not be necessary.

Algorithm 3 Enrollment for Device d do  c_(i) ∈  

_( p), a group element  x = H(c_(i), E, G, p, q)  O = PUF(x)  helper_(i) = O ⊕ ECC(p_(i) ^(priv) mod q)  p_(i) ^(pub) = A_(i) = p_(i) ^(priv) · G mod p  Store {p_(i) ^(pub), c_(i), helper_(i)} end for

Next, PUF-based cryptographic primitives are adapted to secret sharing to permit threshold cryptography founded on PUF or other root of trust. Using the example of an embodiment employing elliptic curve cryptography, distributed key generation is used to generate a number of shares (for example, two: r₁, r₂) of a master private key

^(priv)=(r₁+r₂)mod q), which itself does not need to be generated or constructed at any time during the protocol. The protocol is summarized in Algorithm 4: PUF-DKG, where in an example implementation, (t, n) is chosen as (2, 2).

Algorithm 4 PUF-DKG Goal: Generate shares of master private key

 ^(priv) for 1 ≤ i ≤ n do  Select random polynomial f_(i)(·) = c₀ ^((i)) + . . . + c_(t−1) ^((i))x^(t−1) mod q  Compute f_(i)(j), ∀j ϵ [1, . . . , n]/i  Store coefficient commitments {c_(k) ^((i)) · G mod p}_(kϵ[0, . . . , t−1])  Store share commitments {f_(i)(j) · G mod p}_(jϵ[0, . . . , n])  for 1 ≤ i ≤ n do   Verify ${{f_{i}(j)} \cdot G}\overset{?}{=}{\sum\limits_{k = 0}^{t - 1}{{j^{k}\left( {c_{k}^{(i)} \cdot G} \right)}\mspace{14mu}{mod}\mspace{14mu} p}}$   Verify ${c_{0}^{(i)} \cdot G}\overset{?}{=}{\sum\limits_{i = 1}^{t}{\left( {\left( {\prod\limits_{\underset{j \neq i}{j = 1}}^{t}\;\frac{- j}{i - j}} \right){{f_{i}(j)} \cdot G}} \right)\mspace{14mu}{mod}\mspace{14mu} p}}$  end for  Recover share $r_{i} = {\sum\limits_{j = 1}^{n}{\left( {\left( {\prod\limits_{\underset{j \neq i}{j = 1}}^{t}\;\frac{- j}{i - j}} \right){f_{j}(i)}} \right)\mspace{14mu}{mod}\mspace{14mu} q}}$  Recover public key $\mathcal{P}^{pub} = {{r \cdot G} = {\sum\limits_{j = 1}^{n}{\sum\limits_{i = 1}^{n}{\left( {\left( {\left( {\prod\limits_{\underset{j \neq i}{j = 1}}^{t}\;\frac{- j}{i - j}} \right){f_{j}(i)}} \right) \cdot G} \right)\mspace{14mu}{mod}\mspace{14mu} p}}}}$ end for

Using Algorithms 1 and 2 for storing and retrieving a sensitive value, and Algorithm 4 for performing the initial distributed key generation protocol, arbitrary PUF-enabled threshold cryptographic operations (e.g., decryption, digital signatures, zero knowledge proofs) can be performed. Algorithm 5 describes how to evaluate an arbitrary threshold cryptographic operation

that requires as input a participant's share r_(i). Note that the recovered share r, has already been multiplied by the Lagrange terms

$\left( {\prod\limits_{{j = 1},\;{j \neq i}}^{t}\frac{- j}{i - j}} \right).$

Algorithm 5 PUF-Threshold-OP Goal: Perform threshold operation  

Assume: PUF-DKG (Algorithm 4) has been executed by PUF Device d for Server s do  Issue Command  

  and Auxiliary Information Aux end for for PUF Device d do  for each challenge c_(i) (Challenge c = c₀∥ . . . ∥c_(n)) do   Read challenge c_(i) and helper data h_(i)   Recover share r_(i) ← PUF-Retrieve(c_(i), h_(i))   Perform threshold operation  

 (r_(i), Aux)  end for  Combine threshold operations  

  ← Combine({ 

 (r_(i), Aux)}_(0≤i≤n))  return Result  

end for for Server s do  Process operation  

end for

This enables any threshold cryptographic operation (e.g., decryption, digital signature generation, zero knowledge proofs) to be performed by a PUF-enabled participant without having to generate, reconstruct, or store their private key. Further, from an external perspective (e.g., the server), the PUF-enabled device simply implements standard public key cryptographic protocols. That is, the server does not have to issue a challenge or store helper data, and its interaction with the device is indistinguishable from any standard public key cryptography device. Although threshold cryptography typically considers distributing operations across physically-distinct nodes, in one embodiment of the present invention, threshold cryptography may be applied within a single device. By internalizing the challenge-response functionality of the PUF, and utilizing Algorithms 1 and 2 to locally store and recover a value (e.g., a cryptographic key), arbitrary (e.g., symmetric or asymmetric) cryptographic operations can be performed without need for issuing or storing auxiliary (e.g., challenges or helper data) information.

As an example, a device may be equipped, e.g., with two PUF circuits (e.g., ring oscillator, arbiter, SRAM) and provided with the ability to execute at least two instructions at the same time (e.g., through multiple CPU cores). One embodiment of such a device may comprise a XILINX ARTIX 7 field programmable gate array (FPGA) platform, equipped, e.g., with 215,000 logic cells, 13 Megabytes of block random access memory, and 700 digital signal processing (DSP) slices. In an embodiment employing elliptic curve cryptography, for example, the hardware mathematics engine may be instantiated in the on-board DSP slices, with the PUF construction positioned within the logic cells, and a logical processing core including an input and output to the PUF and constructed to control those and the device's external input and output and to perform algorithms (sending elliptic curve and other mathematical calculations to the math engine) such as those described above. The FPGA may have one or more PUF circuits implemented in separate areas of the FPGA fabric. Simultaneous execution may be accomplished by instantiating multiple software CPUs, e.g., a MICROBLAZE processor. (It is noted that where it is stated herein that the ‘device’ performs an action, it is implicit that such action is or may be carried out by an appropriately-configured processor in the device unless something different is apparent from the context. It is further intended that the word ‘processor’ is inclusive of multiple discrete processors together performing recited actions, processors with multiple cores, etc.).

As a further example, a device may be equipped, e.g., with a block of phase change memory (PCM), dynamic random access memory (DRAM), or other memory that may be physically reconfigured. One embodiment of such a device may comprise an application-specific integrated circuit (ASIC) configured to interact with the DRAM block. Another embodiment of such a device may comprise a system on a chip (SoC) configured to interact with the DRAM block. Yet another embodiment of such a device may be a generic CPU configured to interact with the DRAM block.

An embodiment of the present invention with only one PUF circuit can execute operations over each share sequentially, rather than querying the multiple PUF circuits in parallel. FIG. 2 illustrates a device equipped with two PUF circuits to enable local threshold cryptographic operations; the device may be, for example, an FPGA with a separate core containing each PUF. The potentially extractable output of a single PUF may then be obviated by constructing a local (2, 2) threshold system with each of the parts p_(i) acting as a distinct participant. For example, each part may select a random challenge, run the enrollment algorithm (Algorithm 3) to generate an asymmetric key pair

p_(i) ^(pub)=p_(i) ^(priv)

and locally store its public enrollment information and then together run the distributed key generation protocol (Algorithm 4) and perform all cryptographic operations over a private key that may never actually be constructed. When threshold cryptography is applied within a single device, it may not be necessary to run the enrollment algorithm (Algorithm 3) to generate an asymmetric key pair as all computations are performed internal to the device.

Algorithm 6 describes how a dual-PUF device can compute cryptographic operations in a threshold manner by constructing a (2, 2) threshold sharing within the device using distributed key generation. That is, the two parts establish a private key known to neither part through distributed key generation and publicize the corresponding public key

^(pub). All operations targeted at the device are now performed in a threshold manner through internal collaboration (with each part retrieving its share r_(i) and performing a local threshold operation, and the results are combined to complete a threshold operation

, while the input/output behavior of the device remains unchanged to external systems.

Algorithm 6 Dual-PUF-Threshold-OP Goal: Perform threshold operation  

  at time τ One-Time Setup Stage for each PUF Core p_(i) ∈  

 do  Run Algorithm 3: Enrollment, Publicize p_(i) ^(pub) end for Run (2, 2)-PUF-DKG, Publicize  

^( pub) Evaluation Stage for each PUF Core p_(i) ∈  

  do  Recover share r_(i) ^((τ)) ←PUF-Retrieve(c_(i) ^((τ)) , helper_(i) ^((τ)))  

  ←  

 (r_(i) ^((τ))), PUF core local threshold share end for return  

  ← Combine({ 

 , 

 })

Thus, rather than being constrained to a mapping between a challenge issued to the device and its response (which to an extent may be a function of the challenge), a multi-PUF device d_(i) can have a single static external identity, p_(i) ^(pub). The challenge-response functionality of each PUF core is used to maintain each share of the device's private identity, p_(i) ^(priv), which does not need to be generated or constructed. Each part retrieves its share r_(i) ^((r)) and performs a local threshold operation, and the shares are combined to complete the operation

.

Referring to FIG. 3 and FIG. 4, the operations of an example embodiment employing elliptic curve cryptography, division of a key into two shares, and a (2, 2) threshold operation, are described, via example implementation.

-   -   Enrollment Command 1: During the initial enrollment process, the         server and device agree on an elliptic curve E defined over a         finite field         _(p) and base point G of order q, where p is λ bits long. The         server issues the enrollment command to the device.     -   Distributed Key Generation 2: The device performs a distributed         key generation locally, creating shares (r₀, r₁) of the master         private key (which is never generated or constructed) and its         public key A=(r₀+r₁)·G. Rather than directly add the shares         together (which would construct the private key r=r₀+r₁), the         public key is formed by computing (r₀·G)+(r₁·G).     -   Helper Data Generation 3: The device generates a random         challenge c=c₀∥c₁ where ∥ denotes concatenation and each c_(i)         block is λ bits long. The device links each share r_(i) to the         output O_(i) of the PUF on challenge c_(i) through fuzzy         extraction, which outputs a public helper h_(i). As the PUF         output O_(i) is noisy, there is no guarantee that when queried         on challenge c_(i) in the future, the new output O′_(i) will         satisfy O′_(i)=O_(i). However, it is assumed that O_(i) and         O′_(i) will be t-close with respect to some distance metric         (e.g. Hamming distance). Thus, an error correcting code may be         applied to the PUF output such that at most t errors will still         recover O_(i). Error correction may be applied over each share         r_(i), and this value blinded with the output of the PUF O_(i)         on challenge c_(i), so that each helper value h_(i)=ECC(r_(i))         ⊕O_(i) reveals no information about share r_(i). During recovery         through fuzzy extraction, computing the exclusive-or of         ECC(r_(i)) ⊕O_(i)⊕O′_(i) will return r_(i) whenever O_(i) and         O′_(i) are t-close. The device locally stores the challenge         c=c₀∥c₁ and helper data h=h₀∥h₁, which will allow it to later         recover the shares. Note that both the challenge and the helper         data are public, and reveal nothing about the shares or the         device's private key without invoking the PUF. This process is         described by Algorithm 1.     -   Returned Public Key 4: The device returns its public enrollment         information {A=(r₀+r₁)·G} to the server.     -   Store Enrollment 5: The server stores the device's public         enrollment information along with a (non-sensitive) identifier         that is unique to the device (e.g., a serial number).     -   Threshold Operation Query 6: When the server wishes the device         to perform a cryptographic operation (e.g., decryption, digital         signature generation, zero knowledge proof authentication), it         issues:         -   the appropriate command for the operation to be performed     -   any auxiliary data Aux for the operation (e.g., ciphertext to be         decrypted, a message to be signed)     -   PUF Retrieval 7: The device reads the challenge c=c₀∥c₁ and         helper data h=h₀∥h₁ from its local storage. The device then         queries the PUF on each challenge block c_(i), and combines the         output O′_(i) with the helper block h_(i) and error correcting         code to recover each share block r_(i). This process is         described by Algorithm 2.     -   Threshold Operation 8: The device performs a threshold operation         (r_(i), Aux) over each share T_(i). Algorithm 5 describes this         process for any arbitrary threshold operation         .     -   Combined Threshold Operations 9: The device combines the         threshold operations to form the complete operation         and returns the result to the server.     -   Process Operation 10: The server finally performs any additional         processing required for the operation (e.g., verifying a zero         knowledge proof).         Share Refreshing

Various share refresh protocols allow each of a set of players p_(i)∈

to refresh their share r_(i) ^((τ)) of an original secret r at time period T into a new share r_(i) ^((τ+1)) such that the resulting set of new shares {r_(i) ^((τ+1))}_(i∈[1 . . . n)] remains a sharing of the original secret. This protocol does not require reconstruction of the master secret r, so a mobile adversary would have to compromise t players in a fixed time period τ in order to recover the shared secret. Assuming a polynomial ƒ(⋅) of degree (t−1) represents a shared secret r=ƒ(0) amongst n participants each having a share r_(i)=ƒ(i), and denoting encrypting for player p_(j) as ENC_(j)(⋅) and decryption by p_(j) as DEC_(j)(⋅), the set of players p_(i)∈

can refresh their sharing of r using such a protocol as follows:

-   -   Each player p_(i) defines a new polynomial of degree (t−1) such         that δ_(i)(0)=0:         δ_(i)(⋅)=Δ₁ ^((i)) x+ . . . +Δ _(m) ^((i)) x ^(t−1)     -   where the set {Δ_(m) ^((i))}_(m∈[1 . . . t−1]) is chosen         randomly from         .     -   Each player p_(i) computes the following sets:         {ϵ_(im)=Δ_(m) ^((i)) ·G} _(m∈[1 . . . t−1])         {u _(ij)=δ_(i)(j)}_(j∈[1 . . . n])         {e _(ij) =ENC _(j)(u _(ij))}_(j∈[1 . . . n])     -   and broadcasts a verifiable secret sharing VSS_(i) ^((τ))=         i, τ, {ϵ_(im)}, {e_(ij)}         and their signature SIG_(i)(VSS_(i) ^((τ))).     -   ach player p_(i) recovers u_(ji)=DEC_(i)(ENC_(i)(u_(ji))) and         verifies ∀j≠i:

$\begin{matrix} {{u_{ji} \cdot G}\overset{?}{=}{\sum\limits_{k = 1}^{t - 1}{i^{k}\epsilon_{jk}}}} \\ {= {\sum\limits_{k = 1}^{t - 1}\left( {i^{k}{\Delta_{k}^{(j)} \cdot G}} \right)}} \\ {= {G \cdot {\sum\limits_{k = 1}^{t - 1}{i^{k}\Delta_{k}^{(j)}}}}} \\ {= {{G \cdot {\delta_{j}(i)}} = {G \cdot u_{ji}}}} \end{matrix}$

-   -   Finally, each player p_(i) updates their share from time period         (τ) as:

$\left. r_{i}^{({\tau + 1})}\leftarrow{r_{i}^{(\tau)} + \left( {\sum\limits_{j = 1}^{n}{u_{ji}\;{mod}\; q}} \right)} \right.$ Thus, the refreshed set of shares {r_(i) ^((τ+1))}_(i∈[1 . . . n]) remains a sharing of the master private key

^(priv), and yet knowledge of t−1 or fewer shares from time period τ is useless in time period τ+1.

As outlined in Algorithm 7, participants can update their share r_(i) ^((τ)) in time period τ to a new share r_(i) ^((τ+1)) in the next time period such that the set of shares {r_(i)}_(i∈[1 . . . n]) remains a sharing of the master private key

^(priv).

Algorithm 7 PUF-Share-Update Goal: Generate new share r_(i) ^((τ+1)) for all Participants p_(i) ϵ

 do  Recover share r_(i) ^((τ)) r_(i) ^((τ)) ← PUF-Retrieve(c_(i) ^((τ)), helper_(i) ^((τ)))  Select random polynomial of degree (t − 1) such that δ_(i)(0) = 0: δ_(i)(·) = Δ₁ ^((i))x + . . . + Δ_(m) ^((i))x^(t−1)  Compute {ϵ_(im) = Δ_(m) ^((i)) · G}_(mϵ[1 . . . t−1]) {u_(ij) = δ_(i)(j)}_(jϵ[1 . . . n]) {e_(ij) = ENC_(j)(u_(ij))}_(jϵ[1 . . . n])  Broadcast VSS_(i) ^((τ)) =

i,τ,{ϵ_(im)},{e_(ij)}

, SIG_(i)(VSS_(i) ^((τ)))  Verify ∀j ≠ i ${{{DEC}_{i}\left( {{ENC}_{i}\left( u_{ji} \right)} \right)} \cdot G}\overset{?}{=}{\sum\limits_{k = 1}^{t}{i^{k}\epsilon_{jk}}}$  Update share as: $\left. r_{i}^{({\tau + 1})}\leftarrow{r_{i}^{(\tau)} + \left( {\sum\limits_{j = 1}^{n}{u_{ji}\mspace{14mu}{mod}\mspace{14mu} q}} \right)} \right.$  Store r_(i) ^((τ+1)) and update PUF challenge: {c_(i) ^((τ+1)), helper_(i) ^((τ+1))} ← PUF-Store(r_(i) ^((τ+1))) end for

The hardware device performs Algorithm 7 at Share Refresh 11 in FIG. 4 to generate new shares r_(i) ^((τ+1)) for the next time period τ+1. At PUF Refresh and Store 12, the hardware device generates a new challenge c_(i) ^((τ+1)), which will refresh the challenge-helper pair for the next time period. The hardware device uses the new challenge to store the updated share r_(i) ^((τ+1)). Algorithms 5 and 6 are modified to refresh both the threshold shares as well as the challenge-helper pair, with Algorithms 8 and 9, respectively, reflecting the modifications.

Algorithm 8 PUF-Threshold-OP-Refreshing Goal: Perform threshold operation  

Assume: PUF-DKG (Algorithm 4) has been executed by PUF Device d for Server s do  Issue Command  

  and Auxiliary Information Aux end for for PUF Device d do  for each challenge c_(i) (Challenge c = c₀∥ . . . ∥c_(n)) do   Read challenge c_(i) and helper data h_(i)   Recover share r_(i) ←PUF-Retrieve(c_(i),h_(i))   Perform threshold operation  

 (r_(i), Aux)  end for  Combine threshold operations  

  ← Combine({ 

 (r_(i), Aux)}_(0≤i≤n))  return Result  

 for each share r_(i) do   Update share           r_(i) ^((τ+1)) ← PUF-Share-Update(r_(i) ^((τ)))   Store r_(i) ^((τ+1)) and update PUF challenge:          {c_(i) ^((τ+1)),helper_(i) ^((τ+1))} ← PUF-Store(r_(i) ^((τ+1)))  end for end for for Server s do  Process operation  

end for

Algorithm 9 Dual-PUF-Threshold-OP-Refreshing Goal: Perform threshold operation  

  at time τ One-Time Setup Stage for each PUF Core p_(i) ∈  

  do  Run Algorithm 3: Enrollment, Publicize p_(i) ^(pub) end for Run (2, 2)-PUF-DKG Protocol, Publicize  

^( pub) Evaluation Stage for each PUF Core p_(i) ∈  

  do  Recover share r_(i) ^((τ)) ←PUF-Retrieve(c_(i) ^((τ)) , helper_(i) ^((τ)))  

 ←  

 (r_(i) ^((τ))), PUF core local threshold share  Update share          r_(i) ^((τ+1)) ← PUF-Share-Update(r_(i) ^((τ)))  Store r_(i) ^((τ+1)) and update PUF challenge:         {c_(i) ^((τ+1)) , helper_(i) ^((τ+1))} ← PUF-Store(r_(i) ^((τ+1))) end for return 

  ← Combine({ 

 , 

 })

Referring for example to a single-PUF embodiment as shown in FIG. 1, share updating may optionally be logically split into a preparation phase (Algorithm 10) and an application phase (Algorithm 11). During preparation, each participant generates its random polynomial and distributes its portion of the update to the other participants. After all participants have broadcast their portions of the share update, the preparation phase is complete. (Broadcasting may be omitted if preparation is applied within a single device such as an FPGA).

Algorithm 10 PUF-Share-Update-Preparation Goal: Prepare update to construct share r_(i) ^((τ+1)) for all Participants p_(i) ∈  

  do  Select random polynomial of degree (t − 1) such that δ_(i)(0) = 0:            δ_(i)(·) = Δ₁ ^((i))x + . . . + Δ_(m) ^((i))x^(t−1)  Compute              {ϵ_(im) = Δ_(m) ^((i)) · G}_(mϵ[1...t−1])               {u_(ij) = δ_(i)(j)}_(jϵ[1...n])               {e_(ij) = ENC_(j)(u_(ij))}_(jϵ[1...n])  Broadcast             VSS_(i) ^((τ)) = 

 i, τ, {ϵ_(im)}, {e_(ij)} 

 ,                 SIG_(i)(VSS_(i) ^((τ))) end for

Next, each participant verifies the update information received from other participants and applies the update to its share as set forth in Algorithm 11.

Algorithm 11 PUF-Share-Update-Application Goal: Apply share update to construct r_(i) ^((τ+1)) for all Participants p_(i) ϵ

 do  Recover share r_(i) ^((τ)) r_(i) ^((τ)) ← PUF-Retrieve(c_(i) ^((τ)), helper_(i) ^((τ)))  Verify ∀j ≠ i ${{{DEC}_{i}\left( {{ENC}_{i}\left( u_{ji} \right)} \right)} \cdot G}\overset{?}{=}{\sum\limits_{k = 1}^{t}{i^{k}\epsilon_{jk}}}$  Update share as: $\left. r_{i}^{({\tau + 1})}\leftarrow{r_{i}^{(\tau)} + \left( {\sum\limits_{j = 1}^{n}{u_{ji}\mspace{14mu}{mod}\mspace{14mu} q}} \right)} \right.$ end for

As each threshold operation over a share can be performed independently of the other shares, the device need only recover one share at a time. This process is illustrated in Algorithm 12. Upon receiving a command O and its associated auxiliary information Aux, the device first performs Algorithm 10 to prepare for the share update. Next, the device iteratively performs threshold operations over each share. A share is recovered by reading a challenge-helper pair from non-volatile memory, and using the PUF to regenerate the corresponding share. After performing a threshold operation over the share, the share update is applied using Algorithm 11, which generates the updated share for new time period (τ+1). After computing the threshold operations over each share, the threshold operations are combined to form the result O which is returned to the server.

Algorithm 12 PUF-Threshold-OP-Staggered Goal: Perform threshold operation  

Assume: PUF-DKG (Algorithm 4) has been executed by PUF Device d for Server s do  Issue Command  

  and Auxiliary Information Aux end for for PUF Device d do  for each share r_(i) do   PUF-Share-Update-Preparation  end for  for each challenge c_(i) (Challenge c = c₀∥ . . . ∥c_(n)) do   Read challenge c_(i) and helper data h_(i)   Recover share r_(i) ← PUF-Retrieve(c_(i), h_(i))   Perform threshold operation  

 (r_(i), Aux)   Update share         r_(i) ^((τ+1)) ← PUF-Share-Update-Application(r_(i) ^((τ)))   Store r_(i) ^((τ+1)) and update PUF challenge:          {c_(i) ^((τ+1) ) , helper_(i) ^((τ+1))} ← PUF-Store(r_(i) ^((τ+1)))  end for  Combine threshold operations  

  ← Combine({ 

 (r_(i), Aux)}_(0≤i≤n))  return Result  

end for for Server s do  Process operation  

end for

In one embodiment, a (2, 2) threshold system is constructed internally to the device. Algorithm 13 illustrates an example of a single-PUF (2, 2) threshold construction of the more general Algorithm 12. The device has the share set {r₀, r₁}, and iteratively computes a threshold operation over each share to produce the set {

,

}. Once both threshold operations are complete and the shares have been updated and stored, the two threshold operations are combined into the final output

.

Algorithm 13 Internal-PUF-Threshold-OP-Staggered Goal: Perform threshold operation  

  at time τ One-Time Setup Stage for each Threshold Share r_(i) do  Run Algorithm 3: Enrollment, Publicize p_(i) ^(pub) end for Run (2, 2)-PUF-DKG Protocol, Publicize  

^( pub) Evaluation Stage for each Threshold Share r_(i) do  PUF-Share-Update-Preparation end for for each Threshold Share r_(i) do  Recover share r_(i) ^((τ)) ← PUF-Retrieve(c_(i) ^((τ)) , helper_(i) ^((τ)))   

  ←  

 (r_(i) ^((τ))), Local threshold operation  Update share        r_(i) ^((τ+1)) ← PUF-Share-Update-Application(r_(i) ^((τ)))  Store r_(i) ^((τ+1)) and update PUF challenge:         {c_(i) ^((τ+1)) , helper_(i) ^((τ+1))} ← PUF-Store(r_(i) ^((τ+1))) end for return  

  ← Combine({ 

 , 

 })

The flow of Algorithm 13, a specific single-PUF (2, 2) threshold construction of the more general Algorithm 12, is illustrated in FIG. 5. Prior to Step 1, the share update preparation (Algorithm 10) is performed. In Step 1, the first share r₀ ^(τ) is retrieved and its corresponding local threshold operation is performed. The share update (Algorithm 11) is then applied to r₀ ^(τ) to yield r₀ ^((τ+1)) for the next time period. The updated share is then stored using a new random challenge c₀ ^((τ+1)) which generates the corresponding helper data h₀ ^((τ+1)) which will allow the updated share to be recovered using the PUF. The same process is followed in Step 2 for share r₁ ^(τ). Finally, the combined output

is constructed by combining the two local threshold operations that were performed over each share.

The device has a constant identity

^(pub),

^(priv)

, yet all operations

that require

^(priv) are performed without having to reconstruct

^(priv) and can also be used with values that change after each operation is executed. As each part uses the PUF-Store and PUF-Retrieve algorithms to maintain its share, the (challenge, helper) pair is updated after each operation when PUF-Store is executed. Each share is refreshed for the new time period τ+1, and is stored by generating a new random challenge c_(i) ^((τ+1)) and setting the updated helper to helper_(i) ^((τ+1))←ECC(r_(i) ^((τ+1)))⊕PUF(c_(i) ^((τ+1))). Staggering the threshold operations such that the share regeneration, threshold operation, and share storing occur consecutively (rather than concurrently), precludes the simultaneous recovery of more than one updated share. Any tampering while one share exists would (assuming tampering pushes PUF output beyond error correction limits) prevent recovery of another share, in which case the device cannot perform operations over its private key.

An adversary applying a side channel attack against such an embodiment therefore must extract t or more shares from a period of observation that cannot exceed the period of refreshment. In other words, the adversary must compromise t shares in a given time period τ since any shares from time period τ are useless in time period τ+1. The difficulty of a side channel attack thus can be increased by updating more frequently (even after each operation). (Increasing refresh frequency also may multiply the difficulty inherent in side channel attacks on multiple-PUF device embodiments in which a remote adversary must observe and resolve multiple PUF values simultaneously generated in the device).

In addition to asymmetric operations, symmetric cryptographic operations may also be performed in a threshold manner. Thus all cryptographic operations, asymmetric and symmetric, can be performed over threshold shares rather than the private key. As with the refreshing process described for shares of an asymmetric private key, the shares of a symmetric key may also be refreshed.

Dynamic Membership

The dynamic nature of shares in this construct also permits an embodiment in which the number of participants n participating in a group can be varied dynamically so that participants may join or leave the set of participants in the (t, n) threshold system. In this case, up to n t participants can be removed from the set

simply by leaving them out of the next share refresh protocol. To add a participant p_(j) to the set of participants, each current participant p_(i) generates an extra share u_(ij) from their share update polynomial δ_(i)(⋅).

To add a new participant p_(new) with ID new to the set of participants, their share ƒ(new) must be generated by t members with existing shares. This is performed by each of the t members contributing their share interpolated for p_(new), and blinding these sub-shares with a sharing of zero distributed among the t members. The blinding by a sharing of zero prevents recovery of the t shares r_(i) from the sub-shares. That is, as the t participants are known, distributing a sub-share as

$\left( {r_{i} \cdot \left( {\prod\limits_{\underset{j \neq i}{j = 1}}^{t}\frac{{new} - j}{i - j}} \right)} \right)\mspace{11mu}{mod}\; q$ allows the participant p_(new) to remove the Lagrangian interpolation term

$\left( {\prod\limits_{\underset{j \neq i}{j = 1}}^{t}\frac{{new} - j}{i - j}} \right)$ and recover p_(i)'s share r_(i), as the t members that contribute sub-shares are known to p_(new). To prevent p_(new) from recovering an existing share r_(i) from the sub-share, each sub-share is blinded using separate shares of a sharing of 0 among the t members. Algorithm 14 describes how shares of 0 are generated and distributed to the other existing t members.

Algorithm 14 Share-Blinding Goal: Prepare a sharing of 0 for all Participants p_(i) ∈  

  do  Select random polynomial of degree (t − 1) such that δ_(i)(0) = 0:             δ_(i)(·) = Δ₁ ^((i))x + . . . + Δ_(m) ^((i))x^(t−1)  Compute               {ϵ_(im) = Δ_(m) ^((i)) · G}_(mϵ[1...t−1])                {u_(ij) = δ_(i)(j)}_(jϵ[1...t])                {e_(ij) = ENC_(j)(u_(ij))}_(jϵ[1...t])  Broadcast               VSS_(i) ^((τ)) =  

 i, τ, {ϵ_(im)}, {e_(ij)} 

 ,                   SIG_(i)(VSS_(i) ^((τ))) end for First, each participant p_(i)∈

, |

|≥t generates a polynomial δ_(i)(⋅) where the free coefficient Δ₀ ^((i))=0, and consequently δ_(i)(0)=0. Each participant then distributes shares of their polynomial δ_(i)(j) to the other players p_(j,j≠i)∈

to complete a verifiable sharing of 0. Thus, these shares can be used to blind another sharing without changing the secret being shared.

Upon receiving the sharing of 0, each participant will verify the shares and use them to blind their sub-shares for new participant p_(new). Algorithm 15 describes how the sharing of 0 is verified, the local sub-share of p_(new)'s share is constructed, and how it is blinded before being distributed to p_(new).

Algorithm 15 Share-Construction Goal: Enable new participant p_(new) to construct a new share r_(new) for all Participants p_(i) ϵ

do  Recover share r_(i) ^((τ)) r_(i) ^((τ)) ← PUF-Retrieve(c_(i) ^((τ)), helper_(i) ^((τ)))  Verify ∀j ≠ i ${{{DEC}_{i}\left( {{ENC}_{i}\left( u_{ji} \right)} \right)} \cdot G}\overset{?}{=}{\sum\limits_{k = 1}^{t}{i^{k}\epsilon_{jk}}}$  Construct sub-share r_(new) ^(i) for p_(new) $\left. r_{new}^{i}\leftarrow{\left( {r_{i}^{(\tau)} \cdot \left( {\prod\limits_{\underset{j \neq i}{j = 1}}^{t}\;\frac{{new} - j}{i - j}} \right)} \right)\mspace{14mu}{mod}\mspace{14mu} q} \right.$  Blind sub-share with sharing of 0 from Algorithm 14: Share-Blinding $\left. r_{new}^{i}\leftarrow{r_{new}^{i} + {\sum\limits_{i = 1}^{t}{\left( {u_{ji} \cdot \left( {\prod\limits_{\underset{j \neq i}{j = 1}}^{t}\;\frac{- j}{i - j}} \right)} \right)\mspace{14mu}{mod}\mspace{14mu} q}}} \right.$  p_(new) ← ENC_(p) _(new) (r_(new) ^(i))  Update share r_(i) ^((τ+1)) ← PUF-Share-Update-Application(r_(i) ^((τ)))  Store r_(i) ^((τ+1)) and update PUF challenge: {c_(i) ^((τ+1)), helper_(i) ^((τ+1))} ← PUF-Store(PUF_(i), r_(i) ^((τ+1))) end for for New Participant p_(new) do  for all i ϵ

 do   r_(new) ^(i) ← DEC_(p) _(new) (ENC_(p) _(new) (r_(new) ^(i)))  end for  r_(new) ^((τ)) = Σ_(i=1) ^(t)r_(new) ^(i) mod q  r_(new) ^((τ+1)) ← PUF-Share-Update-Application(r_(new) ^((τ)))  {c_(new) ^((τ+1)), helper_(new) ^((τ+1))} ← PUF-Store(PUF_(new), r_(new) ^((τ+1))) end for First, each of the t participants p_(i)∈

begins by recovering their share r_(i) ^((τ)) and verifying the sharing of 0 they received from Algorithm 14. The sub-share r_(new) ^(i) generated by participant p_(i) is then constructed by performing their portion of Lagrangian interpolation for ƒ(new), where new is the ID of the new participant. This sub-share is subsequently blinded using the sharing of 0 to prevent the new participant p_(new) from recovering the share r_(i) of participant p_(i) by removing the Lagrangian interpolation term. After sending their sub-share to p_(new), participant p_(i) updates their share to the new time period (τ+1) and stores this value using a PUF. All of the received sub-shares are combined by the new participant p_(new) to form their share r_(new) ^((τ)), which is finally updated for time period (τ+1) and stored using a PUF.

In other embodiments, the players may perform the operations of Algorithm 15 in a different order. For example, first each of the t participants p_(i)∈

recover their share r_(i) ^((τ)) and verify the sharing of 0 they received from Algorithm 14. Then p_(i) updates their share to the new time period (τ+1), which is subsequently used to generate the replacement sub-share r_(new) ^(i) by performing their portion of Lagrangian interpolation for ƒ(new), where new is the ID of the new participant. Participant p_(i) stores their updated share r_(i) ^((τ+1)) using the PUF, and sends the sub-share r_(new) ^((τ+1)) to p_(new). All of the received sub-shares are combined by the new participant p_(new) to form their share r_(new) ^((τ+1)), which is then stored using a PUF.

Reconfigurable PUFs

The enforcement of invalidation of used challenge-helper pairs in refreshing-share embodiments of the invention may be improved through the application of reconfigurable PUFs (‘RPUFs’). In one embodiment, an authenticatable device may be provided with a single reconfigurable PUF e.g., a logically-reconfigurable PUF having a reversible configuration process, and e.g., a (2, 2) threshold sharing employed. The PUF configuration is controlled by a parameter, which may be stored locally on the device. Using parameter a to recover one share, a new random parameter b is chosen, the PUF is reconfigured, and the refreshed share is translated into a challenge-helper pair for storage using the PUF configured with parameter b. The PUF is then reconfigured using parameter a to recover the second share, which is subsequently refreshed and translated into a challenge-helper pair for storage using the PUF configured with parameter b. Now, original PUF parameter a is deleted, and the next round will select a new random parameter c to replace parameter b.

In another embodiment, an authenticatable device can be provided with two reconfigurable PUF circuits (e.g., PUF-A, PUF-B) having a non-reversible reconfiguration process, and a (2, 2) threshold sharing employed. After each share is recovered using PUF-A and refreshed, it is translated into a challenge-helper pair for storage using PUF-B. Once both refreshed shares have been stored using PUF-B, the reconfiguration process is applied to PUF-A, such that PUF-A now exhibits a new PUF mapping. The next time the shares are recovered, the same procedure is performed using PUF-B for recovery and PUF-A for commitment.

With reference to FIG. 6, in still another embodiment, an authenticatable device can be provided with a pair of redundant RPUFs (which are preferably irreversibly reconfigurable, such as physically-reconfigurable PUFs) and be configured such that one of the RPUFs (PUF_(1−b), b∈{0, 1}) is randomly selected upon power-up for reconfiguration while the other RPUF's mapping remains unchanged and is used to recover shares from the device's stored challenge-helper values. Parameterizing the preceding PUF-Store and PUF-Retrieve algorithms, Algorithms 16 and 17 now use a specific PUF_(b) to store or retrieve, respectively, a value V_(i).

Algorithm 16 PUF-Store Goal: Store value  

_( i) for PUF Device d do  Select finite field  

_( n) of order n  Select E, an elliptic curve over  

_( n)  Find G ∈ E/ 

_( n), a group generator  for PUF b do   Select challenge c_(i,b) ∈  

_( n) uniformly at random   x = H(c_(i,b), E, G, n)   O = PUF_(b)(x)   helper_(i,b) = P_(i,b) = O ⊕ ECC( 

_( i))   Write {c_(i,b), helper_(i,b)} to non-volatile memory  end for end for

Algorithm 17 PUF-Retrieve Goal: Retrieve value  

_( i) for PUF Device d do  for PUF b do   Read {c_(i,b), helper_(i,b)} from non-volatile memory   x ← H(c_(i,b), E, G, n)   O′ = PUF_(b)(x)    

_( i) ← D((ECC( 

_( i)) ⊕ O) ⊕ O′)  end for end for

Referring again to FIG. 6, at Step 1 the device is powered on and, preferably a source of randomness is used—for example, a true random number generator (‘TRNG’; e.g., FPGA-based such as described in Mazjoobi, “FPGA-based True Random Number Generation using Circuit Metastability with Adaptive Feedback Control,” International Workshop on Cryptographic Hardware and Embedded Systems, Springer Berlin Heidelberg (2011); or IC-based such as FDK Corporation's RPG100 or an ASIC incorporating similar circuitry)—to select a bit b∈{0, 1} uniformly at random. This bit is used to select PUF_(b)∈{PUF₀, PUF₁} for reconfiguration. Then, preferably a source of randomness such as a TRNG is utilized to generate an ephemeral reconfiguration string rc∈{0, 1}^(λ), where λ is the security parameter of the system. The PUF reconfiguration function ƒ(PUF_(1−b), rc)→PUF_(1−b) is applied to PUF_(1−b) using reconfiguration string rc to result in a new PUF_(1−b) mapping, where responses for a fixed challenge c input to PUF_(1−b) and PUF_(1−b) will yield responses t-distant, where t is the error correcting code threshold. Thus, the putative correlated response resulting from the challenge to PUF_(1−b) is not usably correlated to the share due to its reconfiguration from previous state PUF_(1−b). This step need only occur on device power-on, as the current challenge-helper pairs can be maintained in volatile memory (which is preferably protected by the tamper sensitivity of the PUF), and written to non-volatile memory to ensure state recovery in the event of loss of power. That is, once powered on and Step 1 completes, the device need not read the challenge-helper pairs from non-volatile memory, only writing to ensure state recovery in the event of power loss.

As PUF_(1−b) has been reconfigured, the challenge-helper pairs {c_(i,1−b) ^((τ)), h_(i,1−b) ^((τ))} generated using the unmodified PUF_(1−b) will no longer recover the shares, as the PUF reconfiguration function ƒ(

) outputs a new PUF configuration that is t-distant from its argument. Thus, share recovery is performed using PUF_(b), which remains unmodified (FIG. 6 Steps 2-3) and thus its putative responses remain usably correlated to the shares. After the shares r_(i) ^((τ)) have been refreshed to their new representations r_(i) ^((τ+1)), a challenge-helper pair is preferably generated using both PUF_(b) and PUF_(1−b), so that if the device is power-cycled, the shares can be recovered using whichever PUF is not selected for reconfiguration. Finally, the intermediate threshold operations over each share

(r_(i) ^(τ)) are combined into the final cryptographic output

(FIG. 6 Step 4). The boot/reconfiguration operations outlined in FIG. 6 are set forth in pseudo-code in Algorithm 18.

Algorithm 18 Reconfig-Boot Goal: Perform threshold operation  

Assume: PUF-DKG has been executed by PUF Device d for Server s do  Issue Command  

  and Auxiliary Information Aux end for for Device d do  Power On  b ∈ {0, 1} ← TRNG  rc ∈ {0, 1}^(λ) ← TRNG  PUF_(1−b) ← reconfig(PUF_(1−b), rc)  for each share r_(i) ^((τ)) do   PUF-Share-Update-Preparation  end for  for each challenge c_(i,b) ^((τ)) (Challenge c = c_(0,b) ^((τ))∥ . . . ∥c_(n,b) ^((τ))) do   Read challenge c_(i,b) ^((τ)) and helper data h_(i,b) ^((τ))   Recover share r_(i) ^((τ)) ←PUF-Retrieve(PUF_(b),c_(i,b) ^((τ)), h_(i,b) ^((τ)))   Perform threshold operation  

 (r_(i) ^((τ)) , Aux)   Update share         r_(i) ^((τ+1)) ← PUF-Share-Update-Application(r_(i) ^((τ)))   Store r_(i) ^((τ+1)) and update PUF challenge:        {c_(i,b) ^((τ+1)) , helper_(i,b) ^((τ+1))} ← PUF-Store(PUF_(b), r_(i) ^((τ+1)))       {c_(i,1−b) ^((τ+1)) , helper_(i,1−b) ^((τ+1))} ← PUF-Store(PUF_(1−b) , r_(i) ^((τ+1)))  end for  Combine threshold operations  

  ← Combine({ 

 (r_(i) ^((τ)) , Aux)}_(0≤i≤n))  return Result  

  end for for Server s do  Process operation  

  end for

The TRNG and variables not stored in non-volatile memory are preferably protected by the tamper sensitivity property of the PUF, so that an adversary cannot bias the TRNG or alter the bit b selected on power-on. In that regard, reconfigurable PUFs have been demonstrated as a viable protection mechanism for non-volatile memory (see, e.g., Kursawe et al., “Reconfigurable Physical Unclonable Functions Enabling technology for tamper-resistant storage.”).

It is noted that if the device of the foregoing embodiment loses power before storing in memory updated challenge-helper pairs using both the unmodified and reconfigured PUF, when it is powered up the unmodified PUF may be selected for reconfiguration and the shares will be unrecoverable. In an embodiment configured to preclude that possibility, a pair of backup PUFs may be employed, with a set of corresponding challenge-helper pairs generated for each of the backup PUFs. If the primary pair of PUFs are unable to regenerate the shares (e.g., the device performs a test by comparing its regenerated public key against the original stored in non-volatile memory), the backup pair can be invoked. The same general approach that is used for the primary PUF pair can be followed, where the device randomly reconfigures one of the backup PUFs before attempting share regeneration.

More specifically, on power-up, the device proceeds as in FIG. 6 Step 1. After regenerating the shares (which may be incorrect if both primary PUFs have been reconfigured) the device regenerates its putative public key and compares it against a copy stored in non-volatile memory (generated during the original distributed key generation process). If the putative regenerated public key and the stored public key differ (indicating the share recovery may have failed due to a benign power-cycle event that resulted in the reconfiguration of both primary PUFs), the device initiates the backup protocol and invokes the backup PUF pair. The TRNG selects a bit b∈E {0, 1} uniformly at random. This bit is used to select backup PUF_(b)∈{PUF₀, PUF₁} for reconfiguration. The TRNG is queried for an ephemeral reconfiguration string rc∈{0, 1}^(λ), where λ is the security parameter of the system. The PUF reconfiguration function ƒ(PUF_(1−b), rc)→PUF_(1−b) is applied to backup PUF_(1−b) using reconfiguration string rc to result in a new PUF_(1−b) mapping, where responses for a fixed challenge c input to PUF_(1−b) and PUF_(1−b) will yield responses t-distant, where t is the error correcting code threshold.

As backup PUF_(1−b) has been reconfigured, the backup challenge-helper pairs {c_(i,1−b) ^((τ)), h_(i,1−b) ^((τ))} generated using the unmodified backup PUF_(1−b) will no longer recover the shares, as the PUF reconfiguration function ƒ

outputs a new PUF configuration that is t-distant from its argument. Thus, share recovery is performed using backup PUF_(b), which remains unmodified. After the shares r_(i) ^((τ)) have been refreshed to their new representations r_(i) ^((τ+1)), a challenge-helper pair is generated using both backup PUF_(b) and backup PUF_(1−b), as well as primary PUF_(b) and primary PUF_(1−b). This allows the device to return to relying on the primary PUF pair, which were both reconfigured without storing corresponding challenge-helper pairs due to power cycle events. The device has now returned to a state where the primary PUF pair can be successfully invoked on power on, and the backup PUF pair can complete system recovery in the event of a power cycle event that reconfigures both of the primary PUFs. Finally, the intermediate threshold operations over each share

(r_(i) ^(τ)) are combined into the final cryptographic output

.

Physically reconfigurable PUFs (P-RPUFs) can be achieved using phase change memory (PCM), which is a candidate replacement for Flash and DRAM (which itself can be used as a P-RPUF, and, if adopted, would be common to many architectures. A P-RPUF can be instantiated using PCM, and four P-RPUFs can be instantiated on one device by dividing a portion of the memory space into four blocks. Alternately, logically reconfigurable PUFs (L-RPUFs) may be used.

In another embodiment, using a (t, n) sharing in which t<n, one reconfigurable PUF may be allocated per share (rather than allocating any PUF to regenerate all shares). For example, in a (2,4) sharing each of four reconfigurable PUFs would regenerate one of the n=four shares. At power-up, one of the device's PUFs is randomly selected for reconfiguration, resulting in the loss of ability to recover its corresponding share. Upon PUF reconfiguration, however, at least t non-reconfigured PUFs remain to collaborate and help the reconfigured PUF construct a replacement share, as described in Algorithm 19.

Algorithm 19 Reconfig-Boot-One-PUF-Per-Share Goal: Reconfigure a PUF and Generate Replacement Share Assume: PUF-DKG has been executed by PUF Device d for Device d do  Power On  b ∈ {0, 1}^(n) ← TRNG  rc ∈ {0, 1}^(λ) ← TRNG  PUF_(b) ← reconfig(PUF_(b), rc)  Share-Blinding  Share-Construction end for After the selected PUF is reconfigured, the remaining PUFs engage first in Algorithm 14 to construct a sharing of 0, and then engage in Algorithm 15 to construct blinded sub-shares that will enable the reconfigured PUF to construct a replacement share.

This process is illustrated in FIG. 7. Step 1 illustrates the device powering on, preferably randomly selecting a PUF for reconfiguration by choosing a random index b∈{0, . . . , n}, and using a random reconfiguration parameter rc∈{0, 1}^(λ) to reconfigure PUF_(b). In Step 2, a set of at least t PUFs engages in Algorithm 14 to construct a verifiable sharing of 0. In Step 3, each of the t PUFs regenerates its corresponding share r_(i) ^((τ)) and uses it to generate a sub-share r_(b) ^(i) for the reconfigured PUF_(b). This sub-share is blinded using the distributed sharing of 0 performed in Step 2, and the blinded sub-shares are sent to PUF_(b). Each of the (at least) t shares r_(i) ^((τ)) is refreshed for the next time period to r_(i) ^((τ+1)) and subsequently stored using the corresponding PUF_(i). In Step 4, PUF_(b) combines the sub-shares r_(b) ^(i) to construct its share r_(b) ^((τ)) which is subsequently updated to r_(b) ^((r+1)) for the next time period and stored using the now reconfigured PUF_(i).

As the device may experience benign or adversary-initiated malicious power failures, more than one PUF may be reconfigured. In one embodiment, the error correcting code used as part of the fuzzy extraction process can provide a method for determining whether a PUF has been reconfigured. If the number of errors identified during decoding exceeds the error correction capability of the helper string, it is likely due to the PUF being reconfigured. In another embodiment, each PUF may store a non-sensitive test value along with its corresponding challenge-helper pair. The device can then issue the challenge to the PUF as part of the test sequence, and if the value recovered using the helper string does not match the stored value, the PUF is deemed to have been reconfigured. (The challenge-helper-value set associated with each PUF should be periodically refreshed, so that benign hardware aging does not cause the device to incorrectly assume the PUF has been reconfigured). In yet another embodiment, a challenge-response pair can be stored by the device for each PUF. The device can then test whether the PUF has been reconfigured by issuing the challenge and evaluating the Hamming distance between the PUF's response and the reference response stored in memory.

Algorithm 19 can be performed on device power-on, for example, in order to enforce the physical invalidation of old challenge-helper pairs and subsequently construct a replacement for the share lost. Once the device is powered on and this algorithm completes, the device enters normal operational mode such as the example described by Algorithm 20.

Algorithm 20 One-PUF-Per-Share-Operational Goal: Perform threshold operation  

  Assume: PUF-DKG has been executed by PUF Device d for Server s do  Issue Command  

  and Auxiliary Information Aux end for  for Device d do  for each share r_(i) ^((τ)) do   PUF-Share-Update-Preparation  end for  for all PUF_(i), 1 ≤ i ≤ n do   Read challenge c_(i) ^((τ)) and helper data h_(i) ^((τ))   Recover share r_(i) ^((τ)) ← PUF-Retrieve(PUF_(i), c_(i) ^((τ)) , h_(i) ^((τ)))   Perform threshold operation  

  (r_(i) ^((τ)) , Aux)   Update share         r_(i) ^((τ+1)) ← PUF-Share-Update-Application(r_(i) ^((τ)))   Store r_(i) ^((τ+1)) and update PUF challenge:        {c_(i) ^((τ+1)) , helper_(i) ^((τ+1))} ← PUF-Store(PUF_(i), r_(i) ^((τ+1)))  end for  Combine threshold operations  

  ← Combine({ 

 (r_(i) ^((τ)) , Aux)}_(0≤i≤n))  return Result  

  end for for Server s do  Process operation  

  end for

FIG. 8 illustrates an implementation of some principles associated with Algorithm 20. In Step 1, each PUF, regenerates its corresponding share r_(i) ^((τ)) and performs the threshold operation

(r_(i) ^((τ))). The share is subsequently refreshed for the next time period (τ+1), and stored using PUF₁. In Step 2, the individual threshold operations are combined into the final result

.

As with the embodiment of FIG. 6, this embodiment may employ physically or logically reconfigurable PUFs. This embodiment can help reduce the potential for a benign power failure to result in the device being unable to recover t shares in the case where the power failure (and restart) occurs after a power-up PUF reconfiguration but before the construction of a replacement for the share lost to invalidation has completed (see Algorithm 19). For example, a (t, n) sharing wherein n is at least two more than t (e.g., (2, 4)) recovers with Pr=1 from such a power failure; when the device is restarted, regardless of which PUF is reconfigured, at least t valid PUFs will remain available to regenerate the t shares required for correct operation. Two consecutive such power failures with the same timing could be tolerated with Pr=1 if n is at least three greater than t, and so on. The (t, n) embodiment can alternatively be implemented with each RPUF being associated with all shares.

Although certain illustrative embodiments of the invention have been described herein, it will be apparent to those skilled in the art to which the invention pertains that variations and modifications of the described embodiments may be made without departing from the spirit and scope of the invention. For example, separately or in conjunction with the embodiments herein, a (preferably tamper-proof, e.g., encapsulated etc.) capacitor may be employed to provide enough power for the device to carry out all actions following power-up that are required to successfully refresh the device's shares so as to ensure that an adversary cannot power-cycle the device before its shares are refreshed. It is intended that the invention be limited only to the extent required by the appended claims. 

What is claimed is:
 1. An authenticatable device implementing a threshold sharing of a hardware based identity that comprises a (t, n) threshold sharing of a value, wherein n defines a number of threshold shares of the value and t defines a number of threshold shares needed to recover the value, the authenticatable device comprising: a plurality of reconfigurable physical unclonable functions (“RPUFs”), wherein each of at least two of the plurality of RPUFs is associated with at least one respective share of the threshold sharing of the value; a memory; and at least one processor configured to: reconfigure, randomly, one of the at least two RPUFs after power-up of the device, wherein the reconfiguring invalidates a share associated with the reconfigured RPUF; and generate, without generating the value in the memory, information derived from at least t valid threshold shares to enable one or more cryptographic operations that require the value.
 2. The device of claim 1, wherein the at least one processor is further configured to identify invalid shares.
 3. The device of claim 2, wherein the at least one processor is further configured to construct a valid replacement share for the reconfigured RPUF, based on the at least t valid threshold shares.
 4. The device of claim 3, wherein the at least one processor is further configured to store a challenge and helper value for the valid replacement share.
 5. The device of claim 1, wherein the at least one processor is further configured to enroll the at least two RPUFs in the (t, n) threshold sharing of the value without generating the value in the memory.
 6. The device of claim 1, wherein each of the at least two RPUFs is associated with the at least one respective share of the value by an associated challenge and helper pair.
 7. The device of claim 6, wherein the threshold sharing is constructed where n−t is at least two.
 8. The device of claim 1, wherein the at least one processor is configured to enroll the at least two RPUFs in the (t, n) threshold sharing of the value so t valid shares are required to reconstruct the value, where t is less than n.
 9. The device of claim 1, wherein the at least one processor is further configured to trigger a share refresh responsive to enabling a request to execute the one or more cryptographic operations.
 10. The device of claim 1, wherein the at least one processor is further configured to execute one or more cryptographic operations that require the value.
 11. The device of claim 10, wherein the at least one processor is further configured to generate sub-shares associated with the at least t valid threshold shares and blind the sub-shares.
 12. The device of claim 11, wherein the at least one processor is further configured to generate threshold shares of zero values to blind the sub-shares.
 13. A computer implemented method for threshold sharing of a hardware based identity comprising a (t, n) threshold sharing of a value on an authentication device, wherein n defines a number of threshold shares of the value and t defines a number of threshold shares needed to recover the value, the method comprising: randomly reconfiguring, by at least one processor, one of a plurality of reconfigurable physical unclonable functions (“RPUFs”) responsive to power-up of the device, wherein the reconfiguring invalidates a share associated with the reconfigured PUF; and generating, by the at least one processor without generating the value in memory, information derived from at least t valid threshold shares to enable one or more cryptographic operations that require the value.
 14. The method of claim 13, further comprising identifying, by the at least one processor, invalid shares.
 15. The method of claim 14, further comprising constructing, by the at least one processor, a valid replacement share for an invalid share based on the at least t valid threshold shares.
 16. The method of claim 15, further comprising storing, by the at least one processor, a challenge and helper pair for the valid replacement share.
 17. The method of claim 13, further comprising associating, by the at least one processor, each of at least two of the plurality of the RPUFs with at least one respective share of the value by an associated challenge and helper pair.
 18. The method of claim 13, further comprising triggering, by the at least one processor, a share refresh responsive to enabling a request to execute the one or more cryptographic operations.
 19. The method of claim 13, further comprising executing, by the at least one processor, one or more cryptographic operations that require the value.
 20. A non-transitory computer-readable storage medium storing instructions that, when executed by at least one processor, cause the at least one processor to perform a method for threshold sharing of a hardware based identity comprising a (t, n) threshold sharing of a value on an authentication device, wherein n defines a number of threshold shares of the value and t defines a number of threshold shares needed to recover the value, wherein the method comprises: randomly reconfiguring, by at least one processor, one of a plurality of reconfigurable physical unclonable functions (“RPUFs”) responsive to power-up of the device, wherein the reconfiguring invalidates a share associated with the reconfigured PUF; and generating, by the at least one processor without generating the value in memory, information derived from at least t valid threshold shares to enable one or more cryptographic operations that require the value. 